Abstract: A new simulation based algorithm to approximate prices of path dependent European options is introduced. The algorithm is defined for tree-like approximations to the underlying process and makes extensive use of structural properties of the discrete approximation. We indicate the advantages of the new algorithm in comparison to standard Monte Carlo algorithms. In particular, we prove a probabilistic error bound that compares the quality of both approximations. The algorithm is of general applicability and, for a large class of options, it has the same computational complexity as Monte Carlo.